Up to this point, we have progressed in our study of linear algebra without ever specifying whether the entries of our vectors and matrices are real or complex numbers. Although the examples and exercises presented thus far concern real matrices (i.e., matrices having real entries), all the definitions, propositions and results found in previous lectures are applicable without modification to complex matrices (i.e., matrices whose entries are complex numbers). In fact, if you revise those lectures, you will realize that nowhere, and especially in no proof, it is necessary to assume that a matrix or vector be real. The only caveat is that when we deal with complex matrices, we also need to use complex scalars when taking linear combinations.
In this lecture, we are going to revise some elementary facts about complex numbers. We then show some basic properties of complex matrices and provide some useful definitions.
A complex number is a number that can be written aswhere and are real numbers, called the real and imaginary part of the complex number respectively, andis called imaginary unit.
Imaginary numbers allow to find solutions to equations that have no real solutions. For example, the equationhas no real solution, but it has two imaginary solutions
Real numbers are complex numbers that have zero imaginary part. The latter is often omitted, that is, instead of writing we simply write .
An important concept is that of complex conjugate. Given a complex numberits conjugate, denoted by , is
The algebra of complex numbers is similar to the algebra of real numbers. Given two complex numberswe have the following rules:
Note that conjugation is distributive under addition:and under multiplication:
Complex matrices (and vectors) are matrices whose entries are complex numbers.
Given a matrix , its complex conjugate is the matrix such that that is, the -th entry of is equal to the complex conjugate of the -th entry of , for any and .
Example Define the matrix Then its complex conjugate is
The distributive properties that hold for the conjugation of complex numbers hold also for the conjugation of matrices.
Proposition If and are two matrices, then
We have that for any and , by the distributive property of the conjugation of complex numbers under addition.
Proposition If is matrix and is a matrix, then
We have that for any and , by the distributive property of the conjugation of complex numbers under addition and multiplication.
Proposition If is a matrix and is a scalar, then
We havefor any and , by the definition of multiplication of a matrix by a scalar and by the distributive property of the conjugation of complex numbers under multiplication.
A trivial but useful property is that taking the conjugate of a matrix that has only real entries does not change the matrix. In other words, if has only real entries, then
This is a consequence of the fact that a real number can be seen as a complex number with zero imaginary part. But all that conjugation does is to change the sign of the imaginary part of a complex number. Therefore, a real number is equal to its conjugate.
It often happens in matrix algebra that we need to both transpose and take the complex conjugate of a matrix. A special symbol is used to denote the double operation:and is called the conjugate transpose of . In the definition we have used the fact that the order in which transposition and conjugation are performed is irrelevant: whether the sign of the imaginary part of an entry of is switched before or after moving the entry does not change the final result.
Example Define the matrix Then its conjugate transpose is
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